Since for an arbitrary closed set \( K \)  the function \( \delta _{K}^{\phi } \)  is a
locally-semiconcave viscosity solution of the Eikonal equation, several
properties of the set \( \Sigma ^\phi (K) \)  can be deduced from the extensive
literature available for abstract semi-convex functions or for semi-convex
viscosity solutions of uniformly elliptic Hamilton-Jacobi equations. In
particular \( \Sigma ^\phi (K) \)  can be covered by countably many
\( \mathcal {C}^2 \) -hypersurfaces (see [1]}) and upper bounds on its
Hausdorff measure are known (see [2]}). Lower bounds and results on
the propagation of the non-differentiability points can be obtained from
[3]}, [4]} and [5]}. The topological
properties of the set \( \Sigma ^{\phi }(K) \)  in a Euclidean or Riemannian setting
are studied in [6]}, [7]}, [8]}. Being
locally semi-concave \( \delta _{K}^{\phi } \)  is pointwise twice differentiable at
\(  {L}^{n} \)  almost all points in \( \mathbf {R}^{n} \operatorname{\sim }K \)  by the classical
Alexandrov theorem [9]}. We recall that \(\delta _{K}^{\phi } \)  is pointwise
twice differentiable at \( x \)  if and only if there exists a polynomial function
\( P : \mathbf {R}^{n} \rightarrow \mathbf {R}\)  of degree at most 2 such that
\(P(x) = \delta _{K}^{\phi } (x) \quad \text{and} \quad \lim _{y \rightarrow x}\frac{|\delta _{K}^{\phi }(y) - P(y)|}{|y-x|^2} = 0 \,.\) 
